Suppose it is observed that, on average, 35 people are admitted to the intensive care unit of a particular hospital every week. Let the random variable X represent the number of people admitted to this intensive care unit on any particular day.
a) What is the distribution of X?
b) What is the probability that the number of people admitted to this intensive care unit on a particular day is exactly 5?
c) What is the probability that the number of people admitted to this intensive care unit on a particular day is exactly 2?
d) What is the probability that the number of people admitted to this intensive care unit on a particular day is less than 2?
To answer these questions, we need to understand the distribution of X, which represents the number of people admitted to the intensive care unit on any particular day.
a) The distribution of X can be determined based on the given information. The average number of people admitted per week is 35, so we can assume that this is an average rate of admission. Since there are 7 days in a week, the average number of people admitted per day is 35/7 = 5. Therefore, the distribution of X is a Poisson distribution with a mean of 5.
b) To find the probability that exactly 5 people are admitted on a particular day, we can use the formula for the probability mass function (PMF) of a Poisson distribution. The formula is:
P(X=k) = (e^(-λ) * λ^k) / k!
where λ is the mean of the distribution (in this case, 5) and k is the specific value we are interested in (in this case, 5).
So, substituting the values into the formula, we get:
P(X=5) = (e^(-5) * 5^5) / 5!
You can use a scientific calculator, online calculator, or software like Excel or R to calculate this value.
c) To find the probability that exactly 2 people are admitted on a particular day, we can use the same formula as in part b, but with k = 2.
P(X=2) = (e^(-5) * 5^2) / 2!
Again, you can use a calculator or software to compute the value.
d) To find the probability that the number of people admitted on a particular day is less than 2, we need to sum the probabilities of the individual values less than 2 (0 and 1). We can use the cumulative distribution function (CDF) of the Poisson distribution to calculate this.
P(X<2) = P(X=0) + P(X=1)
Using the PMF formula, we can calculate these individual probabilities for X=0 and X=1, and then add them together to get P(X<2).